Band 9, Heft 1

A list of complex numbers Λ is said to be realizable , if it is the spectrum of a nonnegative matrix. In this paper we provide a new sufficient condition for a given list Λ to be universally realizable (UR), that is, realizable for each possible Jordan canonical form allowed by Λ . Furthermore, the resulting matrix (that is explicity provided) is permutative, meaning that each of its rows is a permutation of the first row. In particular, we show that a real Suleĭmanova spectrum, that is, a list of real numbers having exactly one positive element, is UR by a permutative matrix.

Following the recent work of Zheng et al., in this paper, we first present a new extension Hartfiel’s determinant inequality to multiple positive definite matrices, and then we extend the result to a larger class of matrices, namely, matrices whose numerical ranges are contained in a sector. Our result complements that of Mao.

Let L be the infinite lower triangular Toeplitz matrix with first column ( µ , a 1 , a 2 , ..., a p , a 1 , ..., a p , ...) T and let D be the infinite diagonal matrix whose entries are 1, 2, 3, . . . Let A := L + D be the sum of these two matrices. Bünger and Rump have shown that if p = 2 and certain linear inequalities between the parameters µ , a 1 , a 2 , are satisfied, then the singular values of any finite left upper square submatrix of A can be bounded from below by an expression depending only on those parameters, but not on the matrix size. By extending parts of their reasoning, we show that a similar behaviour should be expected for arbitrary p and a much larger range of values for µ , a 1 , ..., a p . It depends on the asymptotics in µ of the l 2 -norm of certain sequences defined by linear recurrences, in which these parameters enter. We also consider the relevance of the results in a numerical analysis setting and moreover a few selected numerical experiments are presented in order to show that our bounds are accurate in practical computations.

In the third part of his famous 1926 paper ‘Quantisierung als Eigenwertproblem’, Schrödinger came across a certain parametrized family of tridiagonal matrices whose eigenvalues he conjectured. A 1991 paper wrongly suggested that his conjecture is a direct consequence of an 1854 result put forth by Sylvester. Here we recount some of the arguments that led Schrödinger to consider this particular matrix and what might have led to the wrong suggestion. We then give a self-contained elementary (though computational) proof which would have been accessible to Schrödinger. It needs only partial fraction decomposition. We conclude this paper by giving an outline of the connection established in recent decades between orthogonal polynomial systems of the Hahn class and certain tridiagonal matrices with fractional entries. It also allows to prove Schrödinger’s conjecture.

We introduce a connection between a near-term quantum computing device, specifically a Gaussian boson sampler, and the graph isomorphism problem. We propose a scheme where graphs are encoded into quantum states of light, whose properties are then probed with photon-number-resolving detectors. We prove that the probabilities of different photon-detection events in this setup can be combined to give a complete set of graph invariants. Two graphs are isomorphic if and only if their detection probabilities are equivalent. We present additional ways that the measurement probabilities can be combined or coarse-grained to make experimental tests more amenable. We benchmark these methods with numerical simulations on the Titan supercomputer for several graph families: pairs of isospectral nonisomorphic graphs, isospectral regular graphs, and strongly regular graphs.

In this article, we evaluate determinants of “block hook” matrices, which are block matrices consist of hook matrices. In particular, we deduce that the determinant of a block hook matrix factorizes nicely. In addition we give a combinatorial interpretation of the aforesaid factorization property by counting weighted paths in a suitable weighted digraph.

The 3-by- n TP-completable patterns are characterized by identifying the minimal obstructions up to natural symmetries. They are finite in number.

It is well-known that the finite difference discretization of the Laplacian eigenvalue problem − Δu = λu leads to a matrix eigenvalue problem (EVP) Ax = λx where the matrix A is Toeplitz-plus-Hankel. Analytical solutions to tridiagonal matrices with various boundary conditions are given in a recent work of Strang and MacNamara. We generalize the results and develop analytical solutions to certain generalized matrix eigenvalue problems (GEVPs) Ax = λBx which arise from the finite element method (FEM) and isogeometric analysis (IGA). The FEM matrices are corner-overlapped block-diagonal while the IGA matrices are almost Toeplitz-plus-Hankel. In fact, IGA with a correction that results in Toeplitz-plus-Hankel matrices gives a better numerical method. In this paper, we focus on finding the analytical eigenpairs to the GEVPs while developing better numerical methods is our motivation. Analytical solutions are also obtained for some polynomial eigenvalue problems (PEVPs). Lastly, we generalize the eigenvector-eigenvalue identity (rediscovered and coined recently for EVPs) for GEVPs and derive some trigonometric identities.

Given a square matrix A , replacing each of its nonzero entries with the symbol * gives its zero-nonzero pattern . Such a pattern is said to be spectrally arbitrary when it carries essentially no information about the eigenvalues of A . A longstanding open question concerns the smallest possible number of nonzero entries in an n × n spectrally arbitrary pattern. The Generalized 2 n Conjecture states that, for a pattern that meets an appropriate irreducibility condition, this number is 2 n . An example of Shitov shows that this irreducibility is essential; following his technique, we construct a smaller such example. We then develop an appropriate algebraic condition and apply it computationally to show that, for n ≤ 7, the conjecture does hold for ℝ, and that there are essentially only two possible counterexamples over ℂ. Examining these two patterns, we highlight the problem of determining whether or not either is in fact spectrally arbitrary over ℂ. A general method for making this determination for a pattern remains a major goal; we introduce an algebraic tool that may be helpful.

By using combinatorial techniques, we obtain an extension of the matrix-tree theorem for general symmetric M -matrices with no restrictions, this means that we do not have to assume the diagonally dominance hypothesis. We express the group inverse of a symmetric M –matrix in terms of the weight of spanning rooted forests. In fact, we give a combinatorial expression for both the determinant of the considered matrix and the determinant of any submatrix obtained by deleting a row and a column. Moreover, the singular case is obtained as a limit case when certain parameter goes to zero. In particular, we recover some known results regarding trees. As examples that illustrate our results we give the expressions for the Group inverse of any symmetric M -matrix of order two and three. We also consider the case of the cycle C 4 an example of a non-contractible situation topologically different from a tree. Finally, we obtain some relations between combinatorial numbers, such as Horadam, Fibonacci or Pell numbers and the number of spanning rooted trees on a path.

We propose an algorithm, which is based on the method given by Kişi and Özdemir in [Math Commun, 23 (2018) 61], to handle the problem of when a linear combination matrix X=∑i=1mciXiX = \sum\nolimits_{i = 1}^m {{c_i}{X_i}} is a matrix such that its spectrum is a subset of a particular set, where c i , i = 1, 2, ..., m , are nonzero scalars and X i , i = 1, 2, ..., m , are mutually commuting diagonalizable matrices. Besides, Mathematica implementation codes of the algorithm are also provided. The problems of characterizing all situations in which a linear combination of some special matrices, e.g. the matrices that coincide with some of their powers, is also a special matrix can easily be solved via the algorithm by choosing of the spectra of the matrices X and X i , i = 1, 2, ..., m , as subsets of some particular sets. Nine of the open problems in the literature are solved by utilizing the algorithm. The results of the four of them, i.e. cubicity of linear combinations of two commuting cubic matrices, quadripotency of linear combinations of two commuting quadripotent matrices, tripotency of linear combinations of three mutually commuting tripotent matrices, and tripotency of linear combinations of four mutually commuting involutive matrices, are presented explicitly in this work. Due to the length of their presentations, the results of the five of them, i.e. quadraticity of linear combinations of three or four mutually commuting quadratic matrices, cubicity of linear combinations of three mutually commuting cubic matrices, quadripotency of linear combinations of three mutually commuting quadripotent matrices, and tripotency of linear combinations of four mutually commuting tripotent matrices, are given as program outputs only. The results obtained are extensions and/or generalizations of some of the results in the literature.

The distance matrix 𝒟( G ) of a connected graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex vi in G is the sum of the distances from vi to all other vertices and T ( G ) is the diagonal matrix of transmissions of the vertices of the graph. The normalized distance Laplacian, 𝒟 𝒧 ( G ) = I − T ( G ) −1/2 𝒟( G ) T ( G ) −1/2 , is introduced. This is analogous to the normalized Laplacian matrix, 𝒧( G ) = I − D ( G ) −1/2 A ( G ) D ( G ) −1/2 , where D ( G ) is the diagonal matrix of degrees of the vertices of the graph and A ( G ) is the adjacency matrix. Bounds on the spectral radius of 𝒟 𝒧 and connections with the normalized Laplacian matrix are presented. Twin vertices are used to determine eigenvalues of the normalized distance Laplacian. The distance generalized characteristic polynomial is defined and its properties established. Finally, 𝒟 𝒧 -cospectrality and lack thereof are determined for all graphs on 10 and fewer vertices, providing evidence that the normalized distance Laplacian has fewer cospectral pairs than other matrices.

We derive an inequality that includes the largest eigenvalue of the adjacency matrix and walks of an arbitrary length of a signed graph. We also consider certain particular cases.

In this paper, by means of the summation property to the Riordan array, we derive some identities involving generalized harmonic, hyperharmonic and special numbers. For example, for n ≥ 0,∑k=0nBkk!H(n.k,α)=αH(n+1,1,α)-H(n,1,α),\sum\limits_{k = 0}^n {{{{B_k}} \over {k!}}H\left( {n.k,\alpha } \right) = \alpha H\left( {n + 1,1,\alpha } \right) - H\left( {n,1,\alpha } \right)} ,and for n > r ≥ 0, ∑k=rn-1(-1)ks(k,r)r!αkk!Hn-k(α)=(-1)rH(n,r,α),\sum\limits_{k = r}^{n - 1} {{{\left( { - 1} \right)}^k}{{s\left( {k,r} \right)r!} \over {{\alpha ^k}k!}}{H_{n - k}}\left( \alpha \right) = {{\left( { - 1} \right)}^r}H\left( {n,r,\alpha } \right)} , where Bernoulli numbers Bn and Stirling numbers of the first kind s ( n , r ).

Using the recent geometric Parter-Wiener, etc. theorem and related results, it is shown that much of the multiplicity theory developed for real symmetric matrices associated with paths and generalized stars remains valid for combinatorially symmetric matrices over a field. A characterization of generalized stars in the case of combinatorially symmetric matrices is given.

Given a weakly stationary, multivariate time series with absolutely summable autocovariances, asymptotic relation is proved between the eigenvalues of the block Toeplitz matrix of the first n autocovariances and the union of spectra of the spectral density matrices at the n Fourier frequencies, as n → ∞. For the proof, eigenvalues and eigenvectors of block circulant matrices are used. The proved theorem has important consequences as for the analogies between the time and frequency domain calculations. In particular, the complex principal components are used for low-rank approximation of the process; whereas, the block Cholesky decomposition of the block Toeplitz matrix gives rise to dimension reduction within the innovation subspaces. The results are illustrated on a financial time series.

Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived as a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank 1 idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) U is a symmetric unitary matrix if and only if it has the form I − 2 E for a symmetric idempotent E , (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.

A proof of the statement per ( A ∘ B ) ≤ per ( A ) per ( B ) is given for 4 × 4 positive semidefinite real matrices. The proof uses only elementary linear algebra and a rather lengthy series of simple inequalities.

We combine matrix theory and graph theory methods to give a complete characterization of the surjective linear transformations of tropical matrices that preserve the cyclicity index. We show that there are non-surjective linear transformations that preserve the cyclicity index and we leave it open to characterize those.

We introduce most of the concepts for q -Lie algebras in a way independent of the base field K . Again it turns out that we can keep the same Lie algebra with a small modification. We use very similar definitions for all quantities, which means that the proofs are similar. In particular, the quantities solvable, nilpotent, semisimple q -Lie algebra, Weyl group and Weyl chamber are identical with the ordinary case q = 1. The computations of sample q -roots for certain well-known q -Lie groups contain an extra q -addition, and consequently, for most of the quantities which are q -deformed, we add a prefix q in the respective name. Important examples are the q -Cartan subalgebra and the q -Cartan Killing form. We introduce the concept q- homogeneous spaces in a formal way exemplified by the examples SUq(1,1)SOq(2){{S{U_q}\left( {1,1} \right)} \over {S{O_q}\left( 2 \right)}} and SOq(3)SOq(2){{S{O_q}\left( 3 \right)} \over {S{O_q}\left( 2 \right)}} with corresponding q- Lie groups and q -geodesics. By introducing a q -deformed semidirect product, we can define exact sequences of q -Lie groups and some other interesting q -homogeneous spaces. We give an example of the corresponding q -Iwasawa decomposition for SL q (2).

The Seidel energy of a simple graph G is the sum of the absolute values of the eigenvalues of the Seidel matrix of G . In this paper we study the Seidel eigenvalues of complete multipartite graphs and find the exact value of the Seidel energy of the complete multipartite graphs.

The lattice path model suggested by E. Deutsch is derived from ordinary Dyck paths, but with additional down-steps of size −3, −5, −7, . . . . For such paths, we find the generating functions of them, according to length, ending at level i , both, when considering them from left to right and from right to left. The generating functions are intrinsically cubic, and thus (for i = 0) in bijection to various objects, like even trees, ternary trees, etc.

Given a real number a ≥ 1, let K n ( a ) be the set of all n × n unit lower triangular matrices with each element in the interval [− a , a ]. Denoting by λ n (·) the smallest eigenvalue of a given matrix, let c n ( a ) = min {λ n ( YY T ) : Y ∈ K n ( a )}. Then cn(a)\sqrt {{c_n}\left( a \right)} is the smallest singular value in K n ( a ). We find all minimizing matrices. Moreover, we study the asymptotic behavior of c n ( a ) as n → ∞. Finally, replacing [− a , a ] with [ a , b ], a ≤ 0 < b , we present an open question: Can our results be generalized in this extension?

Symmetric matrix classes of bandwidth 2 r + 1 was studied in 1972 through binomial coefficients. In this paper, non-symmetric matrix classes with the binomial coefficients are considered where r + s + 1 is the bandwidth, r is the lower bandwidth and s is the upper bandwidth. Main results for inverse, determinants and norm-infinity of inverse are presented. The binomial coefficients are used for the derivation of results.